1919] RANDOM FLIGHTS IN ONE, TWO, OR THREE DIMENSIONS 621 This argument must appear, very roundabout, if the object were merely to obtain the result for n = 2. The advantage is that it admits of easy extension to the general value of n. To this end we take the last stretch ln and the immediately preceding radius sn^, in place of lz and ^ respectively, and then repeat the operation with ln-i, sn-Z) and so on, until we reach 1% and s1 (= /a). The result is evidently 7 sin roc— rx cos roc sin l^x sin I2x sm.lnx etas - ___...__, ...... (5V) or if we suppose, as for the future we shall do, that the I's are all equal, „ , 7. 2 f°° 7 sin rx — roc cos rx /sin loc\n ,„„. (58) ^ ' ' 7T o • OS This is the chance that the resultant is less than r. For the chance that the resultant lies between r and r + dr, we have, as the coefficient of dr, - rJP 9r f00 clr ~r=^n :~-Bin^sin»te ................... (59) dr 7rlnJQ %n x x } Let us now consider the particular case of n = 3, -when dP, 2r °° dx . . ®......................(60) w// /< (/ j () w In this we have . sin rx sii\3lx — %(3 cos (r — 1) x — 3 cos (r + l)x- cos (r — 31) x •+ cos (r + 3Z) *•}. /-°° ^ And — {cos (r - Z) x — cos (r -f 1} x} Jo x o •sin2 ••2 and in like manner for the second pair of cosines. Thus ~? = ^{2r-3 \r~l\ + \r-Bl\} ..................(61) expresses the complete solution. When _. It will be observed that dPa/dr is itself continuous ; but the next derivative changes suddenly at r = I and r = 3Z from one finite value to another. Next take n =4. From (59) dr - = —n I —r sin rx sin4 Ix, TTt4 J o Xsto deal with their cosines, of which all values are to be regarded